简单理论的一个分离定理
A separation theorem for simple theories
摘要：
本文建立了模型理论的工具，以检测简单的理论之间的复杂性的变化。我们发展了一种划分的概括，称为剪切，它取决于所谓的上下文c。这导致了定义c-超稳定性，一个语法概念，其中包括作为特例的超简单性。我们证明了一个分离定理表明，对于任何可数的上下文c和任何两个理论$T_1$，$T_2$，$T_1$是c-超稳定和$T_2$是c-不超稳定的，并为任意大的$μ$，它可以建立模型的任何理论解释$T_1$和$T_2$，其限制$τ(T_1)$是$μ$ -饱和和其限制$τ(T_2)$是不$\aleph_1$ -饱和。(This表明“c-超稳定”实际上是一条分界线。）证明使用了广义Escherichfeucht-Mostowski模型，并且在此过程中，我们阐明了使用这些技术来实现某些类型，而忽略其他类型。在某种意义上，剪切允许我们研究复杂性的相互作用，复杂性来自简单理论中通常的划分概念，而更组合的复杂性则由一般定义检测到。这项工作的灵感来自我们最近的进展Keisler的顺序，但不使用超滤子，而旨在建立这些类的内模理论。
英文摘要：
This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context c. This leads to defining c-superstability, a syntactical notion, which includes supersimplicity as a special case. We prove a separation theorem showing that for any countable context c and any two theories $T_1$, $T_2$ such that $T_1$ is c-superstable and $T_2$ is c-unsuperstable, and for arbitrarily large $μ$, it is possible to build models of any theory interpreting both $T_1$ and $T_2$ whose restriction to $τ(T_1)$ is $μ$-saturated and whose restriction to $τ(T_2)$ is not $\aleph_1$-saturated. (This suggests "c-superstable" is really a dividing line.) The proof uses generalized Ehrenfeucht-Mostowski models, and along the way, we clarify the use of these techniques to realize certain types while omitting others. In some sense, shearing allows us to study the interaction of complexity coming from the usual notion of dividing in simple theories and the more combinatorial complexity detected by the general definition. This work is inspired by our recent progress on Keisler's order, but does not use ultrafilters, rather aiming to build up the internal model theory of these classes.
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